The vertical motion of the disk can be decomposed into a breathing mode motion and a bending mode motion. The former is defined by a vertical pattern with odd parity in the \(V_Z\)  distribution with respect to \(Z\) , and the latter is a pattern with even parity in the \(Z\) -\(V_Z\)  plane [1]}. Both the breathing and the bending modes have been detected in the solar neighborhood, though with differences between works. [2]} found that the vertical motion of the SDSS-SEGUE([3]}; [4]}) stars resemble that of a breathing mode motion, in which stars above the plane are on average moving upwards, while those below the plane, downwards. [5]} argued that stars both above and below the plane are moving towards the plane, as a compression movement. A more complex structure was detected by [6]}. They found opposite breathing modes inside and outside the solar circle: the \(V_Z\)  field implies a rarefaction at \(R<R_0\) , whereas a contraction at \(R>R_0\) . The inner-disk breathing mode found by [7]} is the inverse of that found by [6]}. Moreover, they detected a bending motion in the range of \(6<R<13\,\mathrm {kpc}\) , which is downward interior to the solar circle and upward at \(R\gtrsim 9\,\mathrm {kpc}\) . The findings of [9]} and [10]} agree with a significant upward bending motion in the outer disk. Using data from RAVE DR5 [11]} and the Tycho-Gaia astrometric solution catalogue [12]}, [13]} claimed that a rarefaction-like breathing mode dominates the \(V_Z\)  pattern inside \(R_0\) , while the pattern of the outer disk is dominated by a downward bending mode within \(|Z|\lesssim 1\,\mathrm {kpc}\) . A similar combination of vertical modes was reported by [14]} based on Gaia DR2 astrometric and line-of-sight velocity information, while a positive bending mode was observed at \(R\approx 13\,\mathrm {kpc}\) . In the work of [15]}, the authors found that both the inner and the outer disks present a combination of breathing and bending mode motions with Gaia DR2: a strong bending mode exists across the disk, the velocity of which changes from negative at \(R<R_0\)  to positive at \(R>R_0\)  similar to the pattern found by [7]}; a relatively weak breathing mode shows a smooth distribution over the disk, with amplitudes highly dependent on the distance to the plane. Recently, [17]} adopted proper motions of stars in the direction of the Galactic anticenter from the Gaia EDR3 ([18]}; [19]}) and detected upward vertical motions in both north and south disks. They found that the \(Z>0\)  stars exhibit faster vertical motions than the \(Z<0\)  stars by up to \(\sim 2\,\mathrm {km\,s}^{-1}\)  at \(R<10\,\mathrm {kpc}\) , while at \(R>10\,\mathrm {kpc}\)  the vertical velocities of \(Z<0\)  stars are larger than those of \(Z>0\)  stars by up to \(\sim 6\,\mathrm {km\,s}^{-1}\) , implying a combination of an alternate breathing mode motion and an upward bending mode motion.
